Mathematics
Mathematics or Math (from Greek μάθημα mathema, “knowledge, study, learning”) is a formal science that, starting off from axioms and following logical reasoning, studies the properties and relations of abstract entities such as quantities(numbers), structure, space and change. Mathematics is used to explain the natural world. Through abstraction and use of logic in reasoning, mathematics evolved basing itself on counting, calculus and measurements with the systematic study of shapes and the movement of physical objects. Mathematics, since its inception, have had a practical purpose. The explanations that were supported by logic appeared for the first time with Hellenic mathematics, specially Euclid`s Elements. Nowadays, mathematics is used by the whole world in form of an essential tool in many fields, which are natural sciences, engineering, medicine and social sciences, and even disciplines that, apparently, are not vinculed to them, such as music, for example. Epistemology and Science The epistemologic and scientific character of mathematics has always been discussed. In practice, mathematics is used to study cuantitative relations, estructures, geometric relations and variable magnitudes. Mathematics search for patterns, formulate new conjectures and try to reach the mathematic truth through rigorous deductions. These allows them to establish axioms and definitions apropiate for an specific end. Some of the classic definitions restricted mathematics on the reasoning of cuantities, although just one part of actual mathematics use numbers, which makes predominant the logical analysis of non-cuantitative abstract constructions. Certain discussion exists If the mathematical objects, such as numbers and dots, really exist or simply come from human imagination. The mathematician Benjamin Peirce defined mathematics as: "the science that points to necessary conclusions". On the other hand, Albert Einstein declared: " when the laws of mathematics refer to reality, they are not exact; when they are exact, they dont refer to reality". In conclusion, we can say that mathematics is independent from experience. The scientific character of mathematics has been discussed because its procedures and results has firmness and inevitability nonexistent in other disciplines like physics, chemistry or biology. So mathematics would be tautologic, infallible and a priori, meanwhile others, like geology or physiology, would be fallible and a posteriori. John Stuart Mill affirmed: Logic is not invented nor created nor observed, but judges. Mathematicians can discover new procedures to solve integrals or theorems, but they show incapable of finding a success that puts in doubt the pitagoras theorem or any other, just like it happens with the natural sciences. Mathematics could be understood as a science; If in that case is true, it should be pointed its method and object. Nevertheless, some people pose that mathematics is a formal language, safe, efficent, applicable to the understanding of nature, just like Gallileo said; in addition, much phenomena of social, biological or geological character can be studied through the application of differential equations, calculus of probabilities or set theory. Precisely, the advance of physics and chemistry have required the invention of new concepts, instruments and methods of mathematics, most importantly in the real analysis, complex analysis and matricial analysis. Branches of Mathematics The American Mathematical Society (AMS) recognizes 5000 different branches of mathematics. A wide subdivision of mathematics distinguishes four objects of basic study: quantity, structure, space and change that are corresponded to arithmetics, algebra, geometry and calculus. Also, there are branches of mathematics connected to other fields such as logic, set theory and applied math. Pure Math Quantity: * Natural Numbers, N'': 1, 2, 3 * Integers, ''Z: -3, -2, -1, 0, 1, 2, 3 * Rational Numbers, Q'': 3/2, -1, 5.25 * Irrational Numbers, ''I: π, e, √15 * Real Numbers, R'': -3, √2, 0, 1, π * Complex Numbers, ''C: i'', -9+8''i, ''2''e'i'''''4π/3 Structure: 480px-Rubik's_cube.svg.png|Group Theory 313px-Lattice_of_the_divisibility_of_60.svg.png|Order Theory 376px-Braid-modular-group-cover.svg.png|Algebra 180px-Group_diagdram_D6.svg.png|Graph Theory 225px-Elliptic_curve_simple.svg.png|Number Theory Space: 800px-Mandel_zoom_07_satellite.jpg|Fractal Geometry 500px-Illustration_to_Euclid's_proof_of_the_Pythagorean_theorem.svg.png|Geometry 600px-Hyperbolic_triangle.svg.png|Differential Geometry Measure_illustration.png|Measure Theory Torus.png|Topology Sinusvåg_400px.png|Trigonometry Change: 288px-Conformal_grid_after_Möbius_transformation.svg.png|Complex Analysis 390px-Limitcycle.svg.png|Dynamic Systems 394px-Vector_field.svg.png|Vectorial Calculus 600px-Lorenz_attractor.svg.png|Chaos Theory 682px-Integral_as_region_under_curve.svg.png|Calculus 800px-Airflow-Obstructed-Duct.png|Differential Equations Applied Mathemathics The term "Applied Math" refers to mathematical methods and tools that can be used in the analysis or resolution of problems tjat belong to the area of basic or applied sciences. Statistics and decision sciences Statistics is deals with the techniques of recolection, organization, presetation, analysis of data from number sets and through them and a theoric frame, make inferences take place. Statistics is a fundamental tool for scientific and empiric investigation in the fields of economy, genetics, informatics, engineering, sociology, psycology, medicine, accounting, etc. Computational Mathematics 315px-Simple_feedback_control_loop2.svg.png|Control Theory 420px-Arbitrary-gametree-solved.svg.png|Game Theory 500px-Composite_trapezoidal_rule_illustration_small.svg.png|Numerical Analysis 671px-Two_red_dice_01.svg.png|Probability Theory 790px-BernoullisLawDerivationDiagram.svg.png|Fluid Dynamics 800px-Caesar3.svg.png|Cryptography 800px-GDP_PPP_Per_Capita_IMF_2008.png|Mathematical Economy 800px-Signal_transduction_pathways.svg.png|Biological Mathematics Ch4-structure.png|Chemical Mathematics Gravitation_space_source.png|Mathematical Physics Market_Data_Index_NYA_on_20050726_202628_UTC.png|Financial Mathematics Maximum_boxed.png|Optimization Oldfaithful3.png|Statistics